scholarly journals The outer inverse f(2)T,S of a homomorphism of right R-modules

Filomat ◽  
2019 ◽  
Vol 33 (19) ◽  
pp. 6459-6468
Author(s):  
Zhou Wang
Keyword(s):  
Generalized Inverse ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Drazin Inverse ◽  
Group Inverse ◽  
Outer Inverse ◽  
Basic Properties ◽  
Definition Of ◽  
Necessary And Sufficient

In this paper, we introduce the definition of the generalized inverse f(2)T,S, which is an outer inverse of the homomorphism f of right R-modules with prescribed image T and kernel S. Some basic properties of the generalized inverse f(2)T,S are presented. It is shown that the Drazin inverse, the group inverse and the Moore-Penrose inverse, if they exist, are all the generalized inverse f 2) T,S. In addition, we give necessary and sufficient conditions for the existence of the generalized inverse f(1,2)T,S.

scholarly journals The Generalized Inverse A(2)T, Sof a Matrix Over an Associative Ring

2007 ◽  
Vol 83 (3) ◽  
pp. 423-438 ◽  
Author(s):  
Yaoming Yu ◽  
Guorong Wang
Keyword(s):  
Associative Ring ◽  
Generalized Inverse ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Drazin Inverse ◽  
Group Inverse ◽  
Arbitrary Matrix ◽  
Definition Of ◽  
Necessary And Sufficient ◽  
Explicit Expressions

AbstractIn this paper we establish the definition of the generalized inverse A(2)T, Swhich is a {2} inverse of a matrixAwith prescribed imageTand kernelsover an associative ring, and give necessary and sufficient conditions for the existence of the generalized inverseand some explicit expressions forof a matrix A over an associative ring, which reduce to the group inverse or {1} inverses. In addition, we show that for an arbitrary matrixAover an associative ring, the Drazin inverse Ad, the group inverse Agand the Moore-Penrose inverse. if they exist, are all the generalized inverse A(2)T, S.

scholarly journals Representations of Generalized Inverses and Drazin Inverse of Partitioned Matrix with Banachiewicz-Schur Forms

2016 ◽  
Vol 2016 ◽  
pp. 1-14 ◽  
Author(s):  
Xiaoji Liu ◽  
Hongwei Jin ◽  
Jelena Višnjić
Keyword(s):  
Sufficient Conditions ◽  
Schur Complement ◽  
Block Matrix ◽  
Necessary And Sufficient Conditions ◽  
Generalized Inverses ◽  
Drazin Inverse ◽  
Group Inverse ◽  
Necessary And Sufficient ◽  
Generalized Schur Complement ◽  
Quotient Property

Representations of 1,2,3-inverses, 1,2,4-inverses, and Drazin inverse of a partitioned matrix M=ABCD related to the generalized Schur complement are studied. First, we give the necessary and sufficient conditions under which 1,2,3-inverses, 1,2,4-inverses, and group inverse of a 2×2 block matrix can be represented in the Banachiewicz-Schur forms. Some results from the paper of Cvetković-Ilić, 2009, are generalized. Also, we expressed the quotient property and the first Sylvester identity in terms of the generalized Schur complement.

scholarly journals Generalized Timelike Mannheim Curves in Minkowski Space-Time

2011 ◽  
Vol 2011 ◽  
pp. 1-19 ◽  
Author(s):  
M. Akyig~it ◽  
S. Ersoy ◽  
İ. Özgür ◽  
M. Tosun
Keyword(s):  
Minkowski Space ◽  
Sufficient Conditions ◽  
Space Time ◽  
Necessary And Sufficient Conditions ◽  
Minkowski Space Time ◽  
Mannheim Curve ◽  
Definition Of ◽  
Necessary And Sufficient

We give the definition of generalized timelike Mannheim curve in Minkowski space-time . The necessary and sufficient conditions for the generalized timelike Mannheim curve are obtained. We show some characterizations of generalized Mannheim curve.

Generalized WG inverse

2020 ◽  
pp. 2150072
Author(s):  
Sanzhang Xu ◽  
Hongxing Wang ◽  
Jianlong Chen ◽  
Xiaofeng Chen ◽  
Tiwei Zhao
Keyword(s):  
Generalized Inverse ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Algebraic Form ◽  
Necessary And Sufficient

In this paper, we introduce a generalized inverse of a matrix, namely, the generalized WG inverse, which a generalization of the WG inverse. Several necessary and sufficient conditions such that a matrix to be generalized WG invertible are obtained. Moreover, the formulae of generalized WG inverse of a matrix are given. Finally, we give a algebraic form of the generalized WG inverse.

Fair bets and inductive probabilities

1955 ◽  
Vol 20 (3) ◽  
pp. 263-273 ◽  
Author(s):  
John G. Kemeny
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
The Other ◽  
Inductive Probability ◽  
Other Hand ◽  
Frequency Interpretation ◽  
Definition Of ◽  
Necessary And Sufficient ◽  
Degree Of Confirmation

The question of what constitutes fairness in betting quotients has been studied by Ramsey, deFinetti, and Shimony. Thanks to their combined efforts we now have a satisfactory definition of fairness.On the other hand, the explication of the concept of degree of confirmation (inductive probability) has progressed rapidly in recent years, thanks primarily to Carnap. This explication has usually proceeded by laying down the axioms for frequency-probabilities, and elaborating on these. While in the case where a frequency interpretation is intended these axioms are clearly justified, in our case they have been laid down without any justification. Carnap's presentation has been criticized for just this reason.The purpose of this paper is to show that the probability axioms are necessary and sufficient conditions to assure that the degrees of confirmation form a set of fair betting quotients. In addition it will be shown that one additional, highly controversial, axiom is precisely the condition needed to assure that not only deFinetti's weaker criterion but Shimony's criterion of fairness is also satisfied.

scholarly journals k-Kernel Symmetric Matrices

2009 ◽  
Vol 2009 ◽  
pp. 1-8 ◽  
Author(s):  
A. R. Meenakshi ◽  
D. Jaya Shree
Keyword(s):  
Scalar Product ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Symmetric Matrices ◽  
Fuzzy Matrix ◽  
Basic Properties ◽  
Necessary And Sufficient

In this paper we present equivalent characterizations ofk-Kernel symmetric Matrices. Necessary and sufficient conditions are determined for a matrix to bek-Kernel Symmetric. We give some basic results of kernel symmetric matrices. It is shown that k-symmetric impliesk-Kernel symmetric but the converse need not be true. We derive some basic properties ofk-Kernel symmetric fuzzy matrices. We obtain k-similar and scalar product of a fuzzy matrix.

scholarly journals Conceptualising strategic litigation

2021 ◽  
Author(s):  
Kris van der Pas
Keyword(s):  
Research Method ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Database Research ◽  
Conceptual Relations ◽  
Definition Of ◽  
Necessary And Sufficient ◽  
Exact Definition ◽  
Strategic Litigation

This article conceptualises the term “strategic litigation” in order to provide for a definition of it. Strategic litigation is a tool increasingly used in Europe by individuals and organisations to attain different objectives. Next to that, there is increasing academic attention for the topic. Nevertheless, the exact definition of “strategic litigation” remains unclear. Therefore, this article uses the research method of conceptualisation as well as a database research and additional literature to provide for a definition. It looks firstly at the background concept, involving the range of meanings associated with the term “strategic litigation”, after which a systematised concept is formed. Thereby, use is made of the “necessary and sufficient conditions” (NSC) approach, to develop the conditions necessary and/or sufficient for a case to fit within the category of strategic litigation. Moreover, the external conceptual relations of the term are explored.

scholarly journals Isomorphism of Binary Operations in Differential Geometry

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1634
Author(s):  
Nikita E. Barabanov
Keyword(s):  
Sufficient Conditions ◽  
Canonical Representation ◽  
Necessary And Sufficient Conditions ◽  
Function Parameter ◽  
Binary Operations ◽  
Möbius Addition ◽  
Algebraic Automorphism ◽  
Definition Of ◽  
Necessary And Sufficient ◽  
Metric Tensors

We consider smooth binary operations invariant with respect to unitary transformations that generalize the operations of the Beltrami–Klein and Beltrami–Poincare ball models of hyperbolic geometry, known as Einstein addition and Möbius addition. It is shown that all such operations may be recovered from associated metric tensors that have a canonical form. Necessary and sufficient conditions for canonical metric tensors to generate binary operations are found. A definition of algebraic isomorphism of binary operations is given. Necessary and sufficient conditions for binary operations to be isomorphic are provided. It is proved that every algebraic automorphism gives rise to isomorphism of corresponding gyrogroups. Necessary and sufficient conditions in terms of metric tensors for binary operations to be isomorphic to Euclidean addition are given. The problem of binary operations to be isomorphic to Einstein addition is also solved in terms of necessary and sufficient conditions. We also obtain necessary and sufficient conditions for binary operations having the same function-parameter in the canonical representation of metric tensors to be isomorphic.

On extensions of the Riemann and Lebesgue Integrals by Nets

1975 ◽  
Vol 18 (1) ◽  
pp. 7-17 ◽  
Author(s):  
O. S. Bellamy ◽  
H. W. Ellis
Keyword(s):  
General Theory ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Closed Set ◽  
Denjoy Integral ◽  
Definition Of ◽  
Necessary And Sufficient ◽  
Open Extension

In this note our principal interest is in using nets to give spaces of non-absolutely convergent integrals as extensions of the spaces of absolutely convergent Riemann and Lebesgue integrals. For this purpose we develop a general theory of extensions, by nets, of functions defined on the open intervals with closures in the complement of a fixed closed set, the nets being directed by inclusion for finite disjoint collections of such intervals. Two cases are considered leading to open extension (OE-) and conditional open extension (COE-) nets, the latter being subnets of the former. Necessary and sufficient conditions for the convergence of the OE- and COE-nets are given, those for the COE-nets being similar to conditions that arise in the definition of the restricted Denjoy integral. Properties of inner continuity, weak additivity and the existence of a continuous integral are defined and studied. These relate to the more specialized nets that are suitable for the extension of integrals.

Additivity of the Pn-Integral

1978 ◽  
Vol 30 (4) ◽  
pp. 783-796 ◽  
Author(s):  
George Cross
Keyword(s):  
Sufficient Conditions ◽  
Slight Modification ◽  
Necessary And Sufficient Conditions ◽  
Definition Of ◽  
Necessary And Sufficient

It is known that the Pn-tegral as originally defined is not additive on abutting intervals. This paper offers a slight modification in the definition of the integral and develops necessary and sufficient conditions for the integral to be additive.

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